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Non-Archimedean directed fields \(K(i)\) with o-subfield \(K\) and \(i^2=-1\). (English) Zbl 1307.06013

A linearly ordered field is called non-Archimedean if there is a positive element \(x\) such that \(n1\leq x\) for all positive integers \(n\), where 1 is the identity element of the field. For a non-Archimedean linearly ordered field \(K\), consider the field extension \(K(i)=\{a+bi\mid a,b\in K\}\), where \(i^2=-1\). The paper under review provides a method to construct directed partial orders on \(K(i)\) to make it into a partially ordered algebra over \(K\). The authors call a subset \(V\) of \(K\) a multiplicative segment of \(K\) if \(V\) is a convex additive subgroup of \(K\) and \(1\in V\). For a multiplicative segment \(V\) of \(K\), define \[ P_V=\{a+bi\in K(i)\mid a,b\geq 0\;\&\;(b>0\Rightarrow ab^{-1}\not\in V)\}. \] The main result in the paper shows that \(P_V\) is the positive cone of a directed partial order on \(K(i)\) to make \(K(i)\) into a directed field, and \(P_V\) is unique under certain conditions.

MSC:

06F25 Ordered rings, algebras, modules
12J15 Ordered fields
46A40 Ordered topological linear spaces, vector lattices
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